To determine the correct Pythagorean triple for the distance Elena needs to drive, we can represent the situation as a right triangle where one leg is the distance she drives south (let's call it \( a \)), and the other leg is the distance she drives east (let's call it \( b \)). The hypotenuse of the triangle will represent the straight-line distance from her home to her office, which is 25 miles.
According to the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse (25 miles in this case).
If we want to find integer values that satisfy this equation, we look for Pythagorean triples that can sum to \( 25^2 \) or \( 625 \).
Among the standard Pythagorean triples, we know:
- \( 3^2 + 4^2 = 5^2 \) (i.e., 3, 4, 5)
- \( 5^2 + 12^2 = 13^2 \) (i.e., 5, 12, 13)
- \( 8^2 + 15^2 = 17^2 \) (i.e., 8, 15, 17)
- \( 7^2 + 24^2 = 25^2 \) (i.e., 7, 24, 25)
Based on the above, the values \( 7 \) and \( 24 \) form a valid Pythagorean triple where:
\[ 7^2 + 24^2 = 49 + 576 = 625 = 25^2 \]
So, the correct answer is:
7 miles south and 24 miles east.
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